Kac and new determinants for fractional superconformal algebras.

We derive the Kac and new determinant formulae for an arbitrary (integer) level K fractional superconformal algebra using the BRST cohomology techniques developed in conformal field theory. In particular, we reproduce the Kac determinants for the Virasoro (K = 1) and superconformal (K = 2) algebras. For K ≥ 3 there always exist modules where the Kac determinant factorizes into a product of more fundamental new determinants. Using our results for general K, we sketch the non-unitarity proof for the SU(2) minimal series; as expected, the only unitary models are those already known from the coset construction. We apply the Kac determinant formulae for the spin-4/3 parafermion current algebra (i.e., the K = 4 fractional superconformal algebra) to the recently constructed three-dimensional flat Minkowski space-time representation of the spin-4/3 fractional superstring. We prove the no-ghost theorem for the space-time bosonic sector of this theory; that is, its physical spectrum is free of negative-norm states. 11.17.+y Typeset using REVTEX